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Abstract

We report on a setup for differential x-ray phase-contrast imaging and tomography, that measures the full 2D phase-gradient information. The setup uses a simple one-dimensional x-ray grating interferometer, in which the grating structures of the interferometer are oriented at a tilt angle with respect to the sample rotation axis. In such a configuration, the differential phase images from opposing tomography projections can be combined to yield both components of the gradient vector. We show how the refractive index distribution as well as its x, y, and z gradient components can be reconstructed directly from the recorded projection data. The method can equally well be applied at conventional x-ray tube sources, to analyzer based x-ray imaging or neutron imaging. It is demonstrated with measurements of an x-ray phantom and a rat brain using synchrotron radiation.

The out-of-plane component cancels out because the phase integration, which is implicit in the modified reconstruction kernel, is always performed along the same direction in the camera coordinate system. Considering for instance an integral from left to right through the PMMA cylinder at the bottom of Fig. 2(d), the signal at its left edge is ∂tΦ + ∂zΦ where the signal in the tomographic rotation plane is ∂t Φ > 0 and the out-of-plane signal ∂zΦ < 0. Integrating through the same edge of the sample in Fig. 2(e), the out-of-plane signal has changed sign ∂zΦ > 0, while the in-plane signal has again the same sign as before ∂tΦ > 0. Reconstructing a slice using filtered back projection over a full sample rotation of 2π, each projection pair corresponds to a single line in two-dimensional Fourier space. The out of plane component cancels out since it is once added and once subtracted to this line.

The out-of-plane component cancels out because the phase integration, which is implicit in the modified reconstruction kernel, is always performed along the same direction in the camera coordinate system. Considering for instance an integral from left to right through the PMMA cylinder at the bottom of Fig. 2(d), the signal at its left edge is ∂tΦ + ∂zΦ where the signal in the tomographic rotation plane is ∂t Φ > 0 and the out-of-plane signal ∂zΦ < 0. Integrating through the same edge of the sample in Fig. 2(e), the out-of-plane signal has changed sign ∂zΦ > 0, while the in-plane signal has again the same sign as before ∂tΦ > 0. Reconstructing a slice using filtered back projection over a full sample rotation of 2π, each projection pair corresponds to a single line in two-dimensional Fourier space. The out of plane component cancels out since it is once added and once subtracted to this line.

Figures (4)

Scheme of phase-contrast tomography using a one-dimensional grating interferometer. The sample is mounted on a rotation axis θ to enable tomographic scans. Downstream of the sample are the beam splitter phase grating G1, the absorbing analyzer grating G2 and an imaging detector (not shown here). (a) Conventional arrangement with the grating lines parallel to the sample rotation axis and (b) tilted grating interferometer arrangement with the grating structures tilted, here by an angle of ξ = π/4. The normal nξ to the grating structures (gray arrows) points along the direction of the measured gradient component.

Projections of the x-ray phantom. (a,b) DPC projection pair g0 and gπ recorded with vertical gratings at tomographic projection angles θ = 0 and π respectively. (c) flipped projection (gπ)′, same data as (b) but flipped at the sample rotation axis. (d,e) DPC projection pair recorded with tilted gratings at ξ = π/4 (45°). (f) projection data from (e) flipped at the sample rotation axis. While (a) and (c) only provide the horizontal component of the phase gradient, the projections recorded with tilted gratings (d) and (f) yield two linearly independent components and thus the full 2D gradient vector. All images are displayed with a linear gray scale with deflection angles in the range of [−2.6,2.6] μrad.

Tomographic phase reconstruction of sagittal slices of a rat brain. On the left side are the three components of the reconstructed gradient (a) ∂xδ in horizontal direction, (b) ∂yδ perpendicular to the sagittal plane and (c) ∂zδ in vertical direction. (d) shows the reconstruction of δ using only the gradient components in the tomographic rotation plane, as available in a conventional interferometer setup with vertical gratings. The standard deviation within the white square of constant δ is σδ = 3.9 · 10−9. (e) shows the much cleaner reconstruction of δ obtained by combining the reconstructed gradient components ∂xδ and ∂zδ. Here the standard deviation is σδ = 1.6 · 10−9. All images are displayed with a linear gray scale with the following ranges: (a–c) ∂δ = [−1.5, 1.5] · 10−3 m−1 and (d,e) δ = [3,4] · 10−7.